VIBRATION OF BODIES IN GEXEEAL. 43 



faces, and to compare it with experiment." Only one memoir Mas 

 sent in as a candidate for the prize ; and tins was not crowned, though 

 honorable mention was made of it. 5 The formulae of James Ber- 

 noulli were, according to M. Poisson's statement, defective, in conse- 

 quence of his not taking into account the normal force which acts at 

 the exterior boundary of the plate. 6 The author of the anonymous 

 memoir corrected this error, and calculated the note corresponding to 

 various figures of the nodal lines; and he found an agreement with 

 experiment sufficient to justify his theory. He had not, however, 

 proved his fundamental equation, which M. Poisson demonstrated in 

 a Memoir, read in 1814. 7 At a more recent period also, MM. 

 Poisson and Cauchy (as well as a lady, Mile. Sophie Germain) have 

 applied to this problem the artifices of the most improved analysis. 

 M. Poisson 8 determined the relation of the notes given by the longi- 

 tudinal and the transverse vibrations of a rod ; and solved the problem 

 of vibrating circular plates when the nodal lines are concentric circles. 

 In both these cases, the numerical agreement of his results with expe- 

 rience, seemed to confirm the justice of his fundamental views. 9 He 

 proceeds upon the hypothesis, that elastic bodies are composed of 

 separate particles held together by the attractive forces which they 

 exert upon each other, and distended by the repulsive force of heat. 

 M. Cauchy 10 has also calculated the transverse, longitudinal, and 

 rotatory vibrations of elastic rods, and has obtained results agreeing 

 closely with experiment through a considerable list of comparisons. 

 The combined authority of two profound analysts, as MM. Poisson 

 and Cauchy are, leads us to believe that, for the simpler cases of the 

 vibrations of elastic bodies, Mathematics has executed her task ; but 

 most of the more complex cases remain as yet unsubdued. 



The two brothers, Ernest and William AVeber, made many curious 

 observations on undulations, which are contained in their Wellenlehrf, 

 (Doctrine of Waves,) published at Leipsig in 1825. They were led 

 to suppose, (as Young had suggested at an earlier period,) that 

 Chladni's figures of nodal lines in plates were to be accounted for by 

 the superposition of undulations. 11 Mr. AVheatstone 12 has undertaken 

 to account for Chladni's figures of vibrating square plates by this 



6 Poisson's Him. in Ac. Sc. 1812, p. 169. 8 Ib. p. 220. 



* Ib. 1812, p. 2. 8 Ib. t. viii. 1829. 



9 An. Chim. torn, xxxvi. 1827, p. 90. 10 Exerdces de MatJiematique, iii. ana iv. 



11 Wdlenlehrr, p. 474. 12 Phil. Trans. 1833, p. 59?,. 



